We introduce and study twisted foldings of root systems which generalize usual involutive foldings corresponding to automorphisms of Dynkin diagrams. Our motivating example is the celebrated projection of the root system of type E8 onto the subring of icosians of the quaternion algebra which gives the root system of type H4. Using moment graph techniques we show that such a twisted folding induces a surjective map at the equivariant cohomology level. This is a joint project with Martina Lanini.